In this chapter we examine some simple applications of quantum mechanics in situations where it is possible to model quantum systems accurately by restricting ourselves to a space of states of finite dimension. If each energy level, including degenerate ones, is counted once, the dimension of ℌ is equal to the number of levels, and this is why we use the term system with a finite number of levels . The first two examples (Section 5.1) are taken from quantum chemistry and allow us to study a stationary situation where the Hamiltonian is time-independent. But the most important point in this chapter is the introduction of time dependence, which will be implemented by coupling a two-level system to an external periodic classical field. This will be illustrated by three examples of great practical importance: nuclear magnetic resonance (Section 5.2), the ammonia molecule (Section 5.3), and the two-level atom (Section 5.4). Elementary quantum chemistry The ethylene molecule The ethylene molecule C 2 H 4 will serve as an introduction to the subject. The “skeleton” of this molecule is formed by the so-called σ bonds, pairs of σ electrons of opposite spin common to two carbon atoms or to a carbon and a hydrogen atom, thus forming the (C 2 H 4 ) ++ ion (Fig. 5.1). The remaining two electrons, called π electrons , are mobile – they can jump from one carbon atom to another. It is said that they are delocalized . The separate treatment of the π and σ electrons is, of course, an approximation, but one that plays an important role in the theory of chemical bonding. Let us begin by putting the first π electron in place.
In this chapter we examine some simple applications of quantum mechanics in situations where it is possible to model quantum systems accurately by restricting ourselves to a space of states of finite dimension. If each energy level, including degenerate ones, is counted once, the dimension of ℌ is equal to the number of levels, and this is why we use the term system with a finite number of levels . The first two examples (Section 5.1) are taken from quantum chemistry and allow us to study a stationary situation where the Hamiltonian is time-independent. But the most important point in this chapter is the introduction of time dependence, which will be implemented by coupling a two-level system to an external periodic classical field. This will be illustrated by three examples of great practical importance: nuclear magnetic resonance (Section 5.2), the ammonia molecule (Section 5.3), and the two-level atom (Section 5.4). Elementary quantum chemistry The ethylene molecule The ethylene molecule C 2 H 4 will serve as an introduction to the subject. The “skeleton” of this molecule is formed by the so-called σ bonds, pairs of σ electrons of opposite spin common to two carbon atoms or to a carbon and a hydrogen atom, thus forming the (C 2 H 4 ) ++ ion (Fig. 5.1). The remaining two electrons, called π electrons , are mobile – they can jump from one carbon atom to another. It is said that they are delocalized . The separate treatment of the π and σ electrons is, of course, an approximation, but one that plays an important role in the theory of chemical bonding. Let us begin by putting the first π electron in place.
The book is an introduction to quantum mechanics at a level suitable for the second year in a European university (junior or senior year in an American college). The matrix formulation of quantum mechanics is emphasized throughout, and the student is introduced to Dirac notation from the start. A number of major examples illustrate the workings of quantum mechanics. Several of these examples are taken from solid state physics, with the purpose of showing that quantum mechanics forms the common basis for understanding atoms, molecules and condensed matter. The book contains an introductory chapter which puts the concepts of quantum mechanics into a historical framework. The solid-state applications discussed in this text include the quantum Hall effect, spin waves, quantum wells and energy bands. Other examples feature the two-dimensional harmonic oscillator, coherent states, two-electron atoms, the ammonia molecule and the chemical bond. A large number of homework problems are included.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
This chapter discusses the physical properties of atoms and molecules, which are quantized. This means that they can only take certain discrete values rather than a full range of continuously varying values. The chapter examines the quantized energy levels of a molecule. These are called stationary states, which describe an energy that is independent of time. The chapter also outlines the procedure for constructing a quantum mechanical Hamiltonian, which includes writing down the complete and classical expression for the energy of the system. The chapter reviews the general approach to the determination of energy levels of a physical system. This involves two stages: First, the appropriate Hamiltonian is defined and, in a second step, the eigenfunctions and eigenvalues of the Hamiltonian are determined. The chapter also discusses the Schrödinger equation, which can only be solved for one-body systems.
Quantum mechanics forms the foundation of all modern physics, including atomic, nuclear, and molecular physics, the physics of the elementary particles, condensed matter physics. Modern astrophysics also relies heavily on quantum mechanics. Quantum theory is needed to understand the basis for new materials, new devices, the nature of light coming from stars, the laws which govern the atomic nucleus, and the physics of biological systems. As a result the subject of this book is a required course for most physics graduate students. While there are many books on the subject, this book targets specifically graduate students and it is written with modern advances in various fields in mind. Many examples treated in the various chapters as well as the emphasis of the presentation in the book are designed from the perspective of such problems. For example, the book begins by putting the Schrodinger equation on a spatial discrete lattice and the continuum limit is also discussed, inspired by Hamiltonian lattice gauge theories. The latter and advances in quantum simulations motivated the inclusion of the path integral formulation. This formulation is applied to the imaginary-time evolution operator to project the exact ground state of the harmonic oscillator as is done in quantum simulations. As an example of how to take advantage of symmetry in quantum mechanics, one-dimensional periodic potentials are discussed, inspired by condensed matter physics. Atoms and molecules are discussed within mean-field like treatment (Hartree-Fock) and how to go beyond it. Motivated by the recent intense activity in condensed matter and atomic physics to study the Hubbard model, the electron correlations in the hydrogen molecule are taken into account by solving the two-site Hubbard model analytically. Using the canonical Hamiltonian quantization of quantum electrodynamics, the photons emerge as the quanta of the normal modes, in the same way as the phonons emerge in the treatment of the normal modes of the coupled array of atoms. This is used later to treat the interaction of radiation with atomic matter.
The first objective of this chapter is to briefly review some of the basic ideas about the structure of matter, in particular the concepts of microscopic physics, in order to recall the knowledge gained in previous physics (and chemistry) courses and make it more precise. Our review will be very concise, and most statements will be made without any proof or detailed discussion. A second objective is to give a brief description of some of the crucial stages in the early development of quantum physics. We shall not follow the strict historical order of this development or present the arguments used at the beginning of the last century by the founding fathers of quantum mechanics; rather, we shall stress the concepts which we shall find useful later on. Our last objective is to give an elementary introduction to some of the basic ideas, like those of a quantum particle or energy level, that will reappear throughout this text. We shall base our review on the Bohr theory, which provides a simple, though far from convincing, explanation of how energy levels are quantized and how the spectrum of the hydrogen atom arises. This chapter should be reread later on, once the basic ideas of quantum mechanics have been made explicit and illustrated by examples. From the practical point of view, it is possible to skip the general considerations of Sections 1.1 and 1.2 at the first reading and begin with Section 1.3, returning to those two sections later on as needed.
Abstract The book contains lectures notes for a graduate two-semester course in quantum mechanics. It differs from other quantum mechanics textbooks as various parts of the book are inspired by rather recent advances in various areas of physics. For example, the book begins by putting the Schrödinger equation on a spatial discrete lattice, inspired by Hamiltonian lattice gauge theories (HLGT). The book also discusses the path integral formulation of quantum mechanics and emphasize the adiabatic time evolution in the case of a time-dependent Hamiltonian. As an example of how to use symmetry in quantum mechanics, the book treats one-dimensional periodic potentials. The book also discusses atoms and molecules using mean-field-like treatment, such as the Hartree–Fock approximation, including a discussion on how to go beyond it. Electron–electron correlations in the hydrogen molecule are taken into account with a first quantized formulation of the two-site Hubbard model, which is solved analytically. The book also uses the canonical Hamiltonian quantization of quantum electrodynamics after finding the normal modes, in an analogy with the treatment of the normal modes of an array of atoms, the photons emerge as the quanta of such normal modes, in the same way as the phonons emerge in the treatment of the normal modes of the coupled array of atoms. This Hamiltonian quantization of the electromagnetic field is used later to treat its interaction with atomic matter, without having to follow the usual semiclassical treatment.